On a Four Dimensional Unified Field Theory

The Gravitational, Electromagnetic, Weak & the Strong force are here brought together under a single roof via an extension of Reimann geometry to a new geometry (coined Reimann-Hilbert Space); that unlike Reimann geometry, preserves both the length and the angle of a vector under parallel transport. The affine connection of this new geometry – the Reimann-Hilbert Space, is a tensor and this leads us to a geodesic law that truly upholds the Principle of Relativity. The geodesic law emerging from the General Theory of Relativity (GTR) is well known to be in contempt of the Principle of Relativity which is a principle upon which the GTR is founded. The geodesic law for particles in the GTR must be formulated in special (or privileged) coordinate systems i.e. gaussian coordinate systems whereas the Principle of Relativity clearly forbids the existence of special (or privileged) coordinate systems in manner redolent of the way the Special Theory of Relativity forbids the existence of an absolute (or privileged) frame of reference. In the low energy regime and low spacetime curvature the unified field equations derived herein are seen to reduce to the well known Maxwell-Procca equation, the none-abelian nuclear force field equations, the Lorentz equation of motion for charged particles and the Dirac Equation. Further, to the already existing four known forces, the theory predicts the existence of yet another force. We have coined this the super-force and this force obeys SU(4, 4) gauge invariance. Furthermore, unlike in the GTR, gravitation is here represented by a single scaler potential, and electromagnetic field and the nuclear forces are described by the electromagnetic vector potential (Aμ) which describes the metric tensor i.e. gμν = AμAν. From this (gμν = AμAν), it is seen that gravity waves may not exist in the sense envisaged by the GTR.